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Theorem cmbri 22169
Description: Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
Hypotheses
Ref Expression
pjoml2.1  |-  A  e. 
CH
pjoml2.2  |-  B  e. 
CH
Assertion
Ref Expression
cmbri  |-  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B
) ) ) )

Proof of Theorem cmbri
StepHypRef Expression
1 pjoml2.1 . 2  |-  A  e. 
CH
2 pjoml2.2 . 2  |-  B  e. 
CH
3 cmbr 22163 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B ) ) ) ) )
41, 2, 3mp2an 653 1  |-  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684    i^i cin 3151   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CHcch 21509   _|_cort 21510    vH chj 21513    C_H ccm 21516
This theorem is referenced by:  cmcmlem  22170  cmcm2i  22172  cmbr2i  22175  cmbr3i  22179  pjclem1  22775  pjci  22780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-ov 5861  df-cm 22162
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